Random Fields and Their Applications

In this talk we focus on a general real-valued continuous stationary Gaussian field X characterized by its spectral density |g|², where g is any even real-valued deterministic square integrable function. Our starting point consists in drawing a close connection between critical Besov regularity of the inverse Fourier transform of g and α_X the random pointwise Hölder exponent function of X, which measures local roughness/smoothness of its sample paths at each point. Then, thanks to Littlewood-Paley methods and Hausdorff-Young inequalities, under weak conditions on g, we show that the random function αX is actually a deterministic constant which does not change from point to point. This result means that the field X is of monofractal nature. Also, it is worth mentioning that such a result can easily be extended to the case where X is no longer stationary but has stationary increments.

A multivariate trigonometric regression model is considered. Various discrete modifications of the similar bivariate model received serious attention in the literature on signal and image processing due to multiple applications in the analysis of symmetric textured surfaces. In the presentation asymptotic normality of the least squares estimator for amplitudes and angular frequencies is discussed in multivariate trigonometric model assuming that the random noise is a homogeneous or homogeneous and isotropic Gaussian, in particular, strongly dependent random field on R^M, M>2.

This paper [1] addresses the asymptotic analysis of sojourn functionals of spatiotemporal Gaussian random fields with long-range dependence (LRD) in time also known as long memory. Specifically, reduction theorems are derived for local functionals of nonlinear transformation of such fields, with Hermite rank m≥1, under general covariance structures. These results are proven to hold, in particular, for a family of non–separable covariance structures belonging to Gneiting class. For m=2, under separability of the spatiotemporal covariance function in space and time, the properly normalized Minkowski functional, involving the modulus of a Gaussian random field, converges in distribution to the Rosenblatt type limiting distribution for a suitable range of the long memory parameter. For spatiotemporal isotropic stationary fields on sphere similar results obtained in Marinucci et al. [3]. Some other related results can be found in Makogin and Spodarev [2].

This is joint results with M.D.Ruiz-Medina (Granada University, Spain).

References
[1] Leonenko, N.N. and Ruiz-Medina, M.D. (2022) Sojourn functionals for spatiotemporal Gaussian random fields with Long-memory, Journal of Applied Probability, in press.
[2] Makogin, V. and Spodarev, E. (2022). Limit theorems for excursion sets of subordinated Gaussian random fields with long-range dependence. Stochastics 94, 111–142.
[3] Marinucci, D., Rossi, M. and Vidotto, A. (2020). Non-universal fluctuations of the empirical measure for isotropic stationary fields on S2 × R. Annals of Applied Probability, 31, 2311–2349

The notion of a scalar-valued homogeneous and/or isotropic random field on a Euclidean space or a sphere studied in the classical book by M.I. Yadrenko "Spectral theory of random fields", can be generalized to the vector and tensor-valued case in a nontrivial way. Recently such fields found interesting applications in Continuum Physics and Cosmology. In this presentation, we give a short description of some recent results obtained by the authors in this rapidly growing area.

The investigation of the behaviour for geometric functionals of random fields on manifolds has drawn recently considerable attention. In this paper, we extend this framework by considering fluctuations over time for the level curves of general isotropic Gaussian spherical random fields. We focus on both long and short memory assumptions; in the former case, we show that the fluctuations of u-level curves are dominated by a single component, corresponding to a second-order chaos evaluated on a subset of the multipole components for the random field. We prove the existence of cancellation points where the variance is asymptotically of smaller order; these points do not include the nodal case u = 0, in marked contrast with recent results on the high-frequency behaviour of nodal lines for random eigenfunctions with no temporal dependence. In the short memory case, we show that all chaoses contribute in the limit, no cancellation occurs and a central limit theorem can be established by Fourth-Moment Theorems and a Breuer-Major argument.

Based on a joint paper with Maurizia Rossi and Anna Vidotto.

We will discuss asymptotic properties of multifractal products of random fields. The obtained limit theorems provide sufficient conditions for the convergence of cumulative fields in the spaces L_q. New results on the rate of convergence of cumulative fields will be presented. Simple unified conditions for the limit theorems and the calculation of the Renyi function are given. They are less restrictive than those in the known one-dimensional results. The developed methodology is also applied to multidimensional multifractal measures. Finally, a new class of examples of geometric sub-Gaussian random fields will be presented. In this case, the general assumptions have a simple form and can be expressed in terms of covariance functions only.

The talk is based on the joint results with I. Donhauzer in the manuscript.

We consider the random point process given by the nodal intersections of planar Gaussian random waves and study its scaling limit over growing domains - with specific emphasis on the exact variance asymptotics and functional convergence of its chaotic components. One intriguing result is that the second chaotic component converges to a total disorder field (that is, to a random field containing uncountably many independent standard Gaussian random variables) indexed by closed curves. Such a field already appears in works by Lebowitz (1983) (Coulomb systems) and Buckley and Sodin (2017) (zeros of Gaussian entire functions).

Joint work with M. Notarnicola and A. Vidotto.

The purpose of this lecture is to present general conditions of convergence in distribution for randomly stopped random fields, defined and taking values in metric spaces, and to clarify the connection of these theorems with limit theorems for randomly stopped stochastic processes. The corresponding results and the bibliography of works in the area are presented in [1, 2].

References
[1] Silvestrov, D.S. (2004). Limit Theorems for Randomly Stopped Stochastic Processes. Probability and Its Applications, Springer, London, xvi+398 pp.
[2] Silvestrov, D.S. (2021). Convergence in distribution for randomly stopped random fields. Theor. Probab. Math. Statist., 105, 137-149.

Kriging methods are classically used for the extrapolation of square integrable random fields. For heavy tailed random processes and fields, this theory cannot be applied. We fill this gap and construct a unified approach for the extrapolation of stationary (possibly heavy tailed) random functions using the comparison of their level sets. For that, a new excursion metric for random variables is defined, and its properties are studied. Then it is shown how the new metric is connected to the error-in-measure of level sets of the process and it predictor. For the case of Gaussian processes and linear prediction, the new metric is minimised subject to the constraint that the predictor has the same marginal distribution as the original process. Properties of the new prediction such as consistency and exactness are studied. In the Gaussian case, the existence and uniqueness of the solution of the corresponding linear programming problem with quadratic constraints are analysed. Numerical examples round up the talk.

I will talk about Gaussian isotropic spin random fields on the Riemann sphere, i.e random sections of complex line bundles (the spin weight is half of the degree of the bundle). I will consider those models that exhibit a scaling limit when a parameter grows to infinity. This setting includes sequences of Gaussian pure spin-weighted harmonics, with different asymptotic regimes of the spin and the eigenvalue, that yield different complex fields on the plane as their scaling limit: the Berry (when the spin grows more slowly than the eigenvalue) and the Bargmann-Fock (when the spin grows as fast as the eigenvalue) models, in particular.
I will discuss how the scaling limit determines the leading term of the asymptotic expected geometry in the form of expected Betti numbers and Lipschitz-Killing curvatures of a class of singular loci of such fields including: zeroes, excursion sets, critical points of the norm, and semialgebraic singularities in general.
In particular, the asymptotic of the above quantities is the same for all sequences of Gaussian spin-weighted harmonics for which the spin weight grows more slowly than the eigenvalue.

We present some results on regularity and geometric properties of Gaussian random fields. In particular, we prove four types of limit theorems: the law of the iterated logarithm, uniform modulus of continuity, Chung’s law of the iterated logarithm, and the modulus of nondifferentiability.

For this purpose, we formulate a framework that will be convenient for studying the solutions of stochastic partial differential equations. An important condition in this framework is the property of strong local nondeterminism.